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Math Help - Help with partial diferentiation (chain rule again)

  1. #1
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    Post Help with partial diferentiation (chain rule again)

    The problem I'm working on is for a boundary layer equation for an aerodynamics problem. But I'm having issues working out the partials for it, its been a long while since I've done them and I'm just not getting it to work out. Now for the problem.

    I have functions:
    u(x,y)=Uf^{\prime} where f^{\prime} is \frac{\partial f}{\partial \eta}
    U(x)
    \eta(x,y)=yg(x)

    I need to find the following partials for my problem:
    \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial^{2}u}{\partial y^{2}}, \frac{\partial \eta}{\partial x}, \frac{\partial \eta}{\partial y}, \frac{\partial^{2}\eta}{\partial y^{2}}, \frac{\partial^{2}\eta}{\partial x^{2}}, \frac{\partial^{2}\eta}{\partial x\partial y}

    The second order are the ones I know I'm totally missing, the others I may be getting but it'd be nice to get a sanity check. The ones in terms of \eta and x may cancel out when it goes back into my aerodynamics but I'm not positive of that yet.

    Thanks for any help, believe me I've given this problem a solid 14 hours of head bashing yesterday before asking for help.
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  2. #2
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    Quote Originally Posted by detrepid View Post
    The problem I'm working on is for a boundary layer equation for an aerodynamics problem. But I'm having issues working out the partials for it, its been a long while since I've done them and I'm just not getting it to work out. Now for the problem.

    I have functions:
    u(x,y)=Uf^{\prime} where f^{\prime} is \frac{\partial f}{\partial \eta}
    U(x)
    \eta(x,y)=yg(x)

    I need to find the following partials for my problem:
    \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial^{2}u}{\partial y^{2}}, \frac{\partial \eta}{\partial x}, \frac{\partial \eta}{\partial y}, \frac{\partial^{2}\eta}{\partial y^{2}}, \frac{\partial^{2}\eta}{\partial x^{2}}, \frac{\partial^{2}\eta}{\partial x\partial y}

    The second order are the ones I know I'm totally missing, the others I may be getting but it'd be nice to get a sanity check. The ones in terms of \eta and x may cancel out when it goes back into my aerodynamics but I'm not positive of that yet.

    Thanks for any help, believe me I've given this problem a solid 14 hours of head bashing yesterday before asking for help.
    for the first proceeding formally we get

    \frac{\partial u}{\partial x}=\frac{\partial }{\partial x}\left( Uf' \right) =\frac{\partial U }{\partial x}f'+U\frac{\partial f'}{\partial x}

    By the product rule.

    Now we need to break each of these down.

    \frac{d U }{d x}f'+U\left(\frac{\partial f^2}{\partial^2 \eta} \right)\frac{\partial \eta}{\partial x}=\frac{d U }{d x}f'+U\left(\frac{\partial f^2}{\partial^2 \eta} \right) \left(y\frac{dg}{dx} \right)

    Try this with some of the others
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  3. #3
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    Ok let me post what I've got thus far:
    \frac{\partial u}{\partial x} =\frac{\partial U}{\partial x}f^{\prime}+Uf^{\prime\prime}\frac{\partial \eta}{\partial x}
    \frac{\partial u}{\partial y}=Uf^{\prime\prime}g
    \frac{\partial \eta}{\partial y}=g
    \frac{\partial^{2}\eta}{\partial y^{2}}=0
    \frac{\partial^{2}\eta}{\partial x\partial y}=\frac{\partial g}{\partial x}
    \frac{\partial^{2}u}{\partial y^{2}}=Ug^{2}f^{\prime\prime\prime}

    f^{\prime\prime}=\frac{\partial^{2}f}{\partial\eta  ^{2}}
    f^{\prime\prime\prime}=\frac{\partial^{3}f}{\parti  al\eta^{3}}

    I'm leaving \frac{\partial \eta}{\partial x} as it is because it should cancel out when the correct partials are plugged back into the actual aerodynamic problem.

    That's what I've got on my own, \frac{\partial^{2}u}{\partial y^{2}} is the one I'm pretty sure I'm lost on but I would be I'm probably off on others as well.

    Thanks for the response thus far and any further help.
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